Home›Calculus 1›Limits by factoring›Problem 1.175

Limit of \( \displaystyle \frac{27 x^{3} - 8}{3 x - 2} \) as \( x \to \frac{2}{3} \)

Problem 1.175 · medium

Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{27 x^{3} - 8}{3 x - 2} \).

\[ \lim_{x \to \frac{2}{3}^+}\left(\frac{27 x^{3} - 8}{3 x - 2}\right) \]
limitSet up the limit with the specified direction.✓ Proved
\[ = \lim_{x \to \frac{2}{3}^+}\left(9 x^{2} + 6 x + 4\right) \]
factor cancelFactor the numerator using the difference of cubes formula. Cancel the common factor (3*x - 2) from the numerator and denominator.✓ Proved
\[ = 12 \]
limit algebra simplify simplifyEvaluate the limit by substituting x = 2/3. Simplify the squared term. Perform the multiplication. Final addition.✓ Proved

Answer \( 12 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x - 2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy took the limit from both sides and got the stated value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly factors the numerator, cancels the common term, and evaluates the limit through valid algebraic simplifications. Each step applies a single rule from the allowed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-04: pass — The factoring, cancellation, substitution and arithmetic are all correct and SymPy-verified. Line 5 also evaluates 6*(2/3) while its note mentions only the squared term, which is a trivial, acceptable imprecision.

gpt-oss:20b: dismiss — The middle term was not omitted: 6*(2/3)=4 appears as the second '4' in line 5, and SymPy proved line 5 equals line 4.

Every verdict on record (5)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit through valid algebraic simplifications. Each step applies a single rule from the allowed vocabulary.
gpt-oss:20b: pass 2026-10-04
claude-sonnet-5-5: pass 2026-10-04 — The factoring, cancellation, substitution and arithmetic are all correct and SymPy-verified. Line 5 also evaluates 6*(2/3) while its note mentions only the squared term, which is a trivial, acceptable imprecision.
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator, cancels the common term, and evaluates the limit through standard arithmetic simplification. Each step applies a single rule and is labeled correctly.
gpt-oss:20b: fail (error) 2026-10-04 — In step 5 the term 6*(2/3) was omitted; the correct evaluation of 9*(2/3)^2 + 6*(2/3) + 4 should include that middle term. The missing term leads to an incorrect final value.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-10-04 with SymPy 1.14.0.